cp's OEIS Frontend

A369819 The seventh term of the greedy B_n set of natural numbers.

%I A369819 #14 Feb 29 2024 08:53:25
%S A369819 6,30,124,368,926,2214,4181,8043,13818,23614,34825,54011,84026,109870,
%T A369819 156474,217790,304910,376260,510220,667130,794873,1008048,1302947,
%U A369819 1629264,1916949,2361150,2859694,3467661,3989744,4779270,5479857,6449983,7575912
%N A369819 The seventh term of the greedy B_n set of natural numbers.
%C A369819 Proved in arXiv:2312.10910 that a(n) <=  0.382978*n^5 + O(n^4).
%H A369819 M. B. Nathanson, <a href="https://arxiv.org/abs/2310.14426">The third positive element in the greedy B_h-set</a>, arXiv:2310.14426 [math.NT], 2023.
%H A369819 M. B. Nathanson and Kevin O'Bryant, <a href="https://arxiv.org/abs/2311.14021">The fourth positive element in the greedy B_h-set</a>, arXiv:2311.14021 [math.NT], 2023.
%H A369819 Kevin O'Bryant, <a href="https://arxiv.org/abs/2312.10910">B_h-sets and Rigidity</a>, arXiv:2312.10910 [math.NT], 2023.
%e A369819 a(2) = 30, as all 28 nonincreasing sums from {0,1,3,7,12,20,30}, namely 0+0 < 0+1 < 1+1 < ... < 7+20 < 0+30 < 1+30 < 12+20 <3+30 < 7+30 < 20+20 < 12+30 < 20+30 < 30+30, are distinct, and all other 7-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20,30}.
%o A369819 (Python)
%o A369819 # uses Python code from A369818
%o A369819 from itertools import count, combinations_with_replacement
%o A369819 def A369819(n):
%o A369819     alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1),A369818(n)]
%o A369819     aset = set(sum(d) for d in combinations_with_replacement(alist,n))
%o A369819     blist = []
%o A369819     for i in range(n):
%o A369819         blist.append(set(sum(d) for d in combinations_with_replacement(alist,i)))
%o A369819     for k in count(alist[-1]+1):
%o A369819         for i in range(n):
%o A369819             if any((n-i)*k+d in aset for d in blist[i]):
%o A369819                 break
%o A369819         else:
%o A369819             return k # _Chai Wah Wu_, Feb 28 2024
%Y A369819 Column 7 of A365515.
%K A369819 nonn,more
%O A369819 1,1
%A A369819 _Kevin O'Bryant_, Feb 03 2024